9-1 Two Proportions 447 Alternative Methods: Resampling Methods of Bootstrapping and Randomization The claim that there is no difference in success rates between the E-cigarette group and nicotine replacement group can be tested by using the resampling methods of bootstrapping and randomization. See Section 9-5 “Resampling: Using Technology for Inferences.” Rationale: Why Do the Procedures of This Section Work? Hypothesis Tests With n1p n 1 Ú 5 and n1q n 1 Ú 5, the distribution of p n 1 can be approximated by a normal distribution with mean p1, standard deviation 1p1q1>n1, and variance p1q1>n1 (based on Sections 6-6 and 7-1). This also applies to the second sample. The distributions of pn 1 and p n 2 are each approximated by a normal distribution, so the difference pn 1 - p n 2 will also be approximated by a normal distribution with mean p1 - p2 and variance s 21p1-p22 = s 2 p1 + s 2 p2 = p1q1 n1 + p2q2 n2 (The result above is based on this property: The variance of the differences between two independent random variables is the sum of their individual variances.) The pooled estimate of the common value of p1 and p2 is p = 1x1 + x22>1n1 + n22. If we replace p1 and p2 by p and replace q1 and q2 by q = 1 - p, the variance above leads to the following standard deviation: s1p 1-p22 = Apq n1 + pq n2 We now know that the distribution of pn 1 - p n 2 is approximately normal, with mean p1 - p2 and standard deviation as shown above, so the z test statistic has the form given in the Key Elements box near the beginning of this section. Confidence Interval The form of the confidence interval requires an expression for the variance different from the one given above. When constructing a confidence CP n n n n n n INTERPRETATION The confidence interval limits do not contain 0, suggesting that it is very unlikely that pn 1 - p n 2 = 0, so there is a significant difference between the two proportions. We should therefore reject the null hypothesis of pn 1 = p n 2. It appears that the e-cigarette and nicotine replacement treatment groups have different success rates. YOUR TURN. Do Part (b) of Exercise 7 “Buttered Toast Drop Test.” CAUTION Use of One Confidence Interval Don’t test for equality of two population proportions by determining whether there is an overlap between two individual confidence interval estimates of the two individual population proportions. When compared to the confidence interval estimate of p1 - p2, the analysis of overlap between two individual confidence intervals is more conservative (by rejecting equality less often), and it has less power (because it is less likely to reject p1 - p2 when in reality p1 ≠ p2). See Exercise 25 “Overlap of Confidence Intervals.” Survey Combined with Genetics USA Today reported that among Americans, 53% of women and 26% of men can hear a musical note and match its pitch. That was based on a study conducted by the genetics and biotechnology company 23andMe, named for the fact that a normal human cell has 23 pairs of chromosomes. Two facets of the study are quite interesting: (1) The large difference between the results from women and men, and (2) the fact that the researchers obtained their results by combining survey responses and genetic data from 539,725 customers who did genetic testing.
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